Tagged Questions

Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

learn more… | top users | synonyms

4
votes
1answer
547 views

Exponential Decay of Laplace Coefficients

Laplace coefficients are Fourier coefficients used in Celestial mechanics calculations $$ b^n_s (\alpha) \equiv {1 \over \pi} \int_0^{2\pi} {\cos n \phi \over (1 - 2 \alpha \cos \phi + \alpha^2)^s} d ...
0
votes
0answers
104 views

Complex Multiplications Decomposition Into Lifting Schemes

I have tried searching for explanations for breaking down lifting schemes. The context of this question lies in complex multiplication. The authors have decomposed complex multiplications through ...
1
vote
1answer
332 views

Fourier transform on a simple smooth 1-manifold

Assume a very simple smooth 1-manifold, with a single chart covering, What I'd like to know is, can we use and Fourier transform for functions on this manifold just as we did for the case of ...
6
votes
1answer
424 views

Intuition behind the scaling property of Fourier Transforms

I had a course in PDE last year where we used fourier transforms extensively; I understand the rules of manipulation and can prove the scaling theorem directly from the definition using a ...
5
votes
1answer
536 views

Wavelet Theory — where do I start?

I am in the process of implementing a Fixed-Point Fast Fourier Transform. The Fixed-Point FFT requires mathematical background in the area of wavelets and lifting schemes. What are good ...
2
votes
1answer
165 views

How large are the second, third, fourth, etc. ringing artifacts in Gibbs phenomenon?

I've read that in the Gibbs phenomenon, partial Fourier series will over- or underestimate a function's value in neighborhoods of jump discontinuities. Specifically, the maximum error will converge to ...
0
votes
1answer
475 views

Discrete Fourier Transform Matrix (DFT)

When does it occur that the eigenvalues of a Discrete Fourier Transform matrix are distinct?
3
votes
1answer
358 views

($n$-dimensional) Inverse Fourier transform of $\frac{1}{\| \mathbf{\omega} \|^{2\alpha}}$

I'd like to find the $n$-dimensional inverse Fourier transform of $\frac{1}{\| \mathbf{\omega} \|^{2\alpha}}$ i.e. $$ \int_{\mathbb{R}^n} \frac{1}{ \| \mathbf{\omega} \|^{2\alpha}} e^{2 \pi i ...
4
votes
1answer
204 views

Stuck on complex integral, approximate?

I've been stuck on a particular integral I encountered. I don't need an exact solution, I doubt it even exists. $$f(x)=\frac{e^{-i (r+R-k) x} \left(i-2 e^{i (r+R) x} r x-R x+e^{2 i r x} (R ...
11
votes
1answer
5k views

Criteria for swapping integration and summation order

I have a function (a potential from an electrostatic potential via a Fourier series) in the form of $$V(x, y, z)=\sum_n\sum_m \ a(x, n, m) b(y, n) c(z, m) \int\int f(u, v) d(u,n) e(v,m) du\, dv$$ ...
7
votes
1answer
209 views

Fourier transform of function in $L^{4/3}$

Suppose $f \in L^{4/3}(\mathbb{R}^2)$ and denote its Fourier transform by $\mathscr{F}(f)$. Is it true that the function $g:\mathbb{R}^2 \rightarrow \mathbb{C}$ defined by ...
0
votes
1answer
87 views

A naive question on Haar measure and the module of automorphism

people define haar measure to be left invariant,Weil define module of a automorphism to be the quoient of aX and X,where aX denote X changed under operation “a",if it is left invariant,should module ...
2
votes
1answer
207 views

A calculation involving the normalized area measure

I am reading about the Dirichlet Space right now. The definition of a Dirichlet space is the set of all holomorphic functions in the unit disc that are finite with respect to the semi-norm: $\mid \mid ...
0
votes
3answers
83 views

Simplifying imaginary term of jt in Fourier

I can't figure this out. Don't blame me, but please answer this question. I want to simplify this term: $3(e^{5it} + e^{-5it})$ It would be nice to see a detailed workout. I know the answer is ...
1
vote
1answer
74 views

Is the co-domain of a Hilbert transform of a function the same as the function itself?

Let $f:\mathcal{D}\to\mathcal{D}$ be a function whose domain and co-domain are $\mathcal{D}$. Let $\hat{f}$ be its Hilbert transform, which is defined as $$\hat{f}(t)=\mathcal{H}(f(t))=\frac{1}{\pi} ...
0
votes
2answers
282 views

Is fourier series of a function with $e^{j\theta}$ replaced with a complex variable $z$ holomorphic on the unit disc?

Consider any continuous $2\pi$ periodic function (of bounded variation) $f : \mathbb{R} \to \mathbb{R}$ and its fourier series given as $f(\theta) = \frac{a_o}{2} + \sum\limits_{n = 1}^{\infty} ...
2
votes
1answer
164 views

reference for Fourier series for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$

I am in need of a good reference which has a complete treatment (with all the convergence proofs) for Fourier series representation for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$. ...
0
votes
1answer
4k views

Rebuilding original signal from frequencies, amplitude, and phase obtained after doing an fft

Rebuilding original signal from frequencies, amplitude, and phase obtained after doing an fft. Greetings I'm trying to rebuild a signal from the frequency, amplitude, and phase obtained after I do ...
5
votes
2answers
8k views

FFT bins from exact frequencies

I'm trying to understand a few concepts about Fourier Transforms (mainly in the context of signal processing). Let's suppose a signal is sampled at 10kHz and that the FFT size is 1000. If 1000 ...
2
votes
0answers
99 views

Does the Fourier transform of sequence $f_n\to f$ in $L^2$ converges almost everywhere to $Ff$

$\mathbb K$ is a $\textit{local field}$ if it is any totally disconnected, locally compact, non-discrete, complete field.For examples: $\mathbb Q_p$, any finite extension of $\mathbb Q_p$, field of ...
1
vote
1answer
74 views

Terminology concerning Convergence of Fourier Series

Let $f\in L^1(\mathbb{T})$, and $\sum_{n}a_{n}e^{int}$ its Fourier series. Fix a $t_{0}\in \mathbb{T}$. Suppose $\sum_{n}a_{n}e^{int}$ converges at $t_{0}$. But if it is still possible that ...
3
votes
1answer
99 views

inequality for an integrable real valued function with a compactly supported fourier transform

Let $f$ be an integrable function on $\mathbb{R}$ where support($\hat{f}$) $\subseteq$ [$-\gamma, \gamma$] for some $ 0 < \gamma < 1$ Prove that | $f(x) - f(0)$| $ \leq c \gamma$ |x| ...
2
votes
0answers
249 views

FFT signal post processing

This is more a "post a suggestion" topic rather than a question. And thank you if you are willing to read this whole. I've been studing the code in the Nvidia Cuda SDK regarding how to operate a ...
0
votes
1answer
2k views

Convolution & DFTs: How much zero padding is necessary to avoid circular convolution?

When performing discrete [spatial] convolutions in the frequency domain, how much zero-padding is necessary to avoid the effects of circular convolution? I have a book that almost certainly answers ...
1
vote
1answer
230 views

When has this sequence a triangular shape?

Can someone explain to me why y[n] has a triangular shape? From what I have found, there is a specific range $n_0$ where y[n] is triangular. But right now I have no clue where to start. $^2$ Search ...
2
votes
0answers
100 views

Is the following operator a projection?

Let $P$ be a projection defined on $L_2(\mathbb{R})$ by multiplying with the function of value $1$ for $-1<x<1$ and $0$ otherwise. Let $F$ be the Fourier transform and let $F^{-1}$ be its ...
1
vote
0answers
252 views

Laplace Eigenfunction: Show Eigenvalue is Positive Using Fourier Transform

Problem: Let $ \lambda\in\mathbb{R}, u $ a smooth function, not identically zero, defined on a neighborhood of the unit disc satisfying $ \Delta u+\lambda u = 0 $ in the interior of the unit disc and ...
0
votes
1answer
181 views

Cooley Tukey DFT splitting doubt (should be simple)

I can't understand the basic principle on which the Cooley Tukey algorithm is based, the algorithm says I can split in two parts the DFT computation like in the following $$\begin{matrix} X_k= ...
2
votes
0answers
130 views

Turning real roots into curves (for visualisation)

One can obviously map a set of real numbers $x_1, x_2, \ldots x_N$ to a curve in 2-D via $y=(x-x_1)(x-x_2)\ldots(x-x_N)$. Thinking about data visualisation, one can portray a set of $N$ observations ...
2
votes
2answers
1k views

Proof of Fourier Transform

Where F is the fourier transform, how can you show that $$\mathcal F(x\cdot f(x)) = −i \frac{d\mathcal F}{dw}.$$ I understand that you are meant to apply the inverse transform to the left hand side, ...
1
vote
0answers
143 views

Estimate the Hilbert transform

Let $1\leq p<∞$: Suppose that there exists a constant $C>0$ such that for all $f\in S(\mathbb{R})$ with $L^p$ norm one we have $$\biggl|\{x:|H(f)(x)|>1\}\biggr|\leq C.$$ Here $H(f)$ is ...
1
vote
3answers
181 views

Why is it useful to express PDE solutions as $L^2$-convergent series?

The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the ...
3
votes
1answer
221 views

Proving that the characters of an infinite Abelian group is a basis for the space of functions from the group to $\mathbb{C}$

Let $A$ be an arbitrary (possibly infinite) Abelian group. A character $\chi$ is a group homomorphism from $A$ to the multiplicative group of complex numbers. I can prove that if $A$ is finite then ...
2
votes
1answer
392 views

What is the generalization of Parseval's theorem into spherical coordinates?

what is the relationship between the total power of a function given in spherical coordinates in the Fourier domain: $E_k=\int_{\mathbb{R}^3}|F(k,\Theta,\Phi)|^2k^2 \sin(\Theta)\,dk\,d\Theta\, ...
0
votes
1answer
1k views

Using the Fourier integral theorem to evaluate the improper integrals

I'm trying to brush up with Fourier series with Apostol's Mathematical Analysis. I was looking through the Fourier chapter and its Fourier integral theorem. I'm slightly confused on how to approach it ...
7
votes
3answers
666 views

Decomposing a discrete signal into a sum of rectangle functions

Hello math@stackexchange community ! I have a simple question that seems to have a non trivial answer. Given a discrete one dimensional signal $w(x)$ defined in a finite range, and the boxcar ...
3
votes
0answers
520 views

How do I find the inverse Fourier transform of a function that is separable into a radial and an angular part?

I need to take the inverse Fourier transform of a function that is initially specified in spherical coordinates: $f(r, \theta, \phi) = \int_{R^3}F(k, ...
3
votes
1answer
350 views

How do I find the inverse Hankel transform of $k^2e^{-k^2}$?

I am trying to solve: $f_l(r)=\int_0^{\infty}e^{-k^2}k^4j_l(kr)dk$, where $j_l$ is the spherical Bessel function of the first kind, for any integer l >= 0. Thanks in advance for any answers!
2
votes
1answer
435 views

Which functions are tempered distributions?

Today's problem originates in this conversation with Willie Wong about the Fourier transform of a Gaussian function $$g_{\sigma}(x)=e^{-\sigma \lvert x \rvert^2},\quad x \in \mathbb{R}^n;$$ where ...
8
votes
1answer
552 views

Fourier transform of Schrödinger kernel: how to compute it?

Let $$K_t(x)=\frac{1}{(4 \pi i t)^{\frac{n}{2}}}e^{i \frac{\lvert x \rvert^2}{4t}}\quad x \in \mathbb{R}^n,\ t \in \mathbb{R},\ t\ne 0.$$ Clearly this is not a $L^1$ or $L^2$ function with respect ...
3
votes
0answers
528 views

Fourier transform of vector-valued functions (e.g. differential forms)

Consider $L^2(\mathbb R^n, \mathbb R^m)$. There should be a Fourier transform for these functions, like in the case $L^2( \mathbb R^n, \mathbb R )$. I wonder how these can be defined. The application ...
2
votes
1answer
886 views

How do I find the Fourier transform of a function that is separable into a radial and an angular part?

how do I find the Fourier transform of a function that is separable into a radial and an angular part: $f(r, \theta, \phi)=R(r)A(\theta, \phi)$ ? Thanks in advance for any answers!
0
votes
1answer
696 views

LTI: How to derive the impulse response of this system?

Well, i transform g and x into the frequency domain. u[n] = 1, n ≥ 0 u[n] = 0, n < 0 \begin{aligned} x[n] & = u[n] \\ h_1[n] & = (\frac{1}{2})^n u[n] \\ g[n] & = (\frac{1}{2})^n u[n] \\ ...
2
votes
1answer
182 views

Trouble deriving DE for fourier transform from DE of function

I am trying to derive an equation which is a standard result in physics (the momentum space Schrödinger equation). (Background: The wavefunction is a complex valued function of position coordinates ...
29
votes
2answers
1k views

Explicitly reconstructing a function from its moments

Let $f$ be an integrable real valued function defined on $[0,\infty)$. Let $$m_n=\int_0^\infty f(x)x^n \mathrm dx$$ be the $n^{th}$ moment, and suppose that all of these integrals converge ...
1
vote
1answer
3k views

Fourier basis functions

What are fourier basis functions? And how do I prove that fourier basis functions are orthonormal?
1
vote
1answer
70 views

Help with fourier transforms

I am going through a book and having trouble with reproducing some results mentioned. The aim is to solve for $D_{s}$ from equation (1) below $\int ...
6
votes
2answers
4k views

Why is 8x8 matrix chosen for Discrete Cosine Transform?

In JPEG and MPEG, why is 8x8 matrix chosen for Discrete Cosine Transform? Why not any other, say 64x64?
2
votes
2answers
292 views

Calculating $\displaystyle{\int_0^\infty e^{-i\omega t}dt}$

I was studying Fourier Transform; I could answer to this $$\int_{-\infty}^\infty e^{-i\omega t}dt$$ by Fourier Transform, but I have problem in $$\int_0^\infty e^{-i\omega t}dt.$$ I would be grateful ...
3
votes
3answers
520 views

What is the relationship between different definitions of Fourier transform?

I always see various definitions of Fourier transform. A standard form is: $$\hat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-2\pi ix\cdot\xi}dx$$ and its attached inversion is ...